\latex{\section*{Overview}\addcontentsline{toc}{subsection}{Overview}}


This package contains Java classes providing methods to 
compute mass, density, distribution and complementary
distribution functions for some multi-dimensional discrete
and continuous probability distributions.
% and to perform goodness-of-fit tests. % and collect statistics.  
It does not generate random numbers for multivariate distributions;
for that, see the package \externalclass{umontreal.iro.lecuyer}{randvarmulti}.

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\subsection* {Distributions}

We recall that the {\em distribution function\/} of a {\em continuous\/} random
vector $X= \{x_1, x_2, \ldots, x_d\}$ with {\em density\/} $f(x_1, x_2, \ldots,
x_d)$ over the
 $d$-dimensional space $R^d$  is
\begin{eqnarray}
  F(x_1, x_2, \ldots, x_d) &=& P[X_1\le x_1, X_2\le x_2, \ldots, X_d\le x_d] \\[6pt]
    &=&
  \int_{-\infty}^{x_1}\int_{-\infty}^{x_2} \cdots  \int_{-\infty}^{x_d} f(s_1, s_2,
 \ldots, s_d)\; ds_1 ds_2 \ldots ds_d    \label{eq:FDist}
\end{eqnarray}
while that of a {\em discrete\/} random vector $X$ with {\em mass function\/}
 $\{p_1, p_2, \ldots, p_d\}$ over a fixed set of real numbers is
\begin{eqnarray}
  F(x_1, x_2, \ldots, x_d) &=& P[X_1\le x_1, X_2\le x_2, \ldots, X_d\le x_d] \\[6pt]
   &=& \sum_{i_1\le x_1}\sum_{i_2\le x_2} \cdots
   \sum_{i_d\le x_d} p(x_1, x_2, \ldots, x_d),     \label{eq:FDistDisc}
\end{eqnarray}
where $p(x_1, x_2, \ldots, x_d) = P[X_1 = x_1, X_2 = x_2, \ldots, X_d = x_d]$.
For a discrete distribution over the set of integers, one has
\begin{eqnarray}
  F (x_1, x_2, \ldots, x_d) &=& P[X_1\le x_1, X_2\le x_2, \ldots, X_d\le x_d] \\[6pt]
   &=& \sum_{s_1=-\infty}^{x_1} \sum_{s_2=-\infty}^{x_2} \cdots
   \sum_{s_d=-\infty}^{x_d} p(s_1, s_2, \ldots, s_d),   \label{eq:FDistDiscInt}
\end{eqnarray}
where $p(s_1, s_2, \ldots, s_d) = P[X_1=s_1, X_2=s_2, \ldots, X_d=s_d]$.

We define $\bar{F}$, the {\em complementary distribution function\/} 
of $X$, as
\eq
 \bar{F} (x_1, x_2, \ldots, x_d) = P[X_1\ge x_1, X_2\ge x_2, \ldots, X_d\ge x_d].
\endeq
